Curves and angles between them

Question

How do you define-: (a) Angle between curves (b) Angle between straight line and a curve (c) Angle between tangent and a curve

Answer 1

a) The angle between two curves is measured by finding the angle between their tangents at the point of intersection. b) The angle between a straight line and a curve can be measured by drawing a tangent on curve at the point of intersection of straight line and curve. Then finding angle between tangent and curve. c) find the slope of tangent to the curve. We know Tan A=slope where A is angle between tangent and curve. find A

Answer 2

Note: the angle between two curves is defined for a specific intersection point of the curves (there may be more than one) - different intersection points can have different angles.

---

For all curves $c$ in $\Bbb{R}^n$, let $\partial c(p)$ be the line tangent to $c$ at the point $p$.

Note that $\partial(\partial c(p))=\partial c(p)$ ($\partial$ is idempotent). If $c$ is a straight line, then $\partial c=c$ at every point on $c$ (in other words, a straight line is its own tangent line).

Let $\angle(c_1(p),c_2(p))$ denote the angle between the curves $c_1$ and $c_2$ at the point $p$

The angle between a line and itself is always $0$.

Definition: The angle between two curves is the angle between their tangent lines.

(a) Let $c_1$ and $c_2$ be curves in $\Bbb{R}^n$. $\angle(c_1(p),c_2(p))=\angle(\partial c_1(p),\partial c_2(p))$. (b) Let $l$ be a straight line, and $c$ a curve in $\Bbb{R}^n$. By definition $\partial l=l$, thus $\angle(l(p),c(p))=\angle(\partial l(p),\partial c(p))=\angle(l(p),\partial c(p))$.

(c) the angle between a tangent line $t$ and a curve $c$ is the angle between $t$ and $\partial c(p)$. Note that the line tangent to the tangent line is the tangent line itself, hence $\angle(t(p),c(p))=\angle(\partial t(p),\partial c(p))=\angle(t(p),\partial c(p))$.

(both of the above figures show this)

If $t$ is tangent to $c$ at a point $p$, then, by definition, $t=\partial c(p)$, whence $\angle(t(p),c(p))=\angle(\partial c(p),\partial c(p))=0$

Answer 3

The answer can be also given verbally using line vectors for tangents at the intersection point.

Let the given line be L, the tangent at point of intersection P given curve be T. Then the angle between curve and line is given by dot product

$$ \cos^{-1} \frac {T.L}{|T||L|}$$

Similarly let the given curves be $ C_1,C_2$, let the tangents at point of intersection P of given curves be $T_1,T_2$. Then the angle between the two curves and line is given by dot product

$$ \cos^{-1} \frac {T_1.T_2}{|T_1||T_2|}.$$